FLUCTUATIONS AND SIMPLECHAOTIC DYNAMICS

J.P. CRUTCHFIELD and J.D. FARMER

PhysicsBoard of Studies, UniversityofCalifornia, Santa Cruz, California 95064, U.S.A.

and

B.A. HUBERMAN

Xerox Palo Alto Research Center, Palo Alto, California 94304, U.S.A.

NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM

PHYSICSREPORTS(Review Sectionof PhysicsLetters)92, No. 2 (1982)45—82. North-HollandPublishingCompany

FLUCTUATIONS AND SIMPLE CHAOTIC DYNAMICS

J.P. CRUTCHFIELD and J.D. FARMER*PhysicsBoard of Studies, Universiiy of California, Santa Crux, California 95064, USA.

and

B.A. HUBERMANXerox Palo Alto Research Center, Palo Alto, California 94304, U.S.A.

ReceivedJuly 1982

Contents:

I. Introduction 47 AppendixA. Fluctuationsin a driven anharmonicoscillator 762. Dynamicsin theabsenceof fluctuations 48 Appendix B. Characteristic exponents in period-doubling3. Dynamicsin thepresenceof fluctuations 57 regimes 774. Noiseasadisorderingfield 68 AppendixC. Equivalenceof parametricandadditivenoise 805. Concludingremarks 74 References 81

Abstract:We describetheeffectsof fluctuationson theperiod-doublingbifurcationto chaos.We studythedynamicsof mapsof the interval in theabsence

of noiseand numericallyverify the scalingbehaviorof the Lyapunovcharacteristicexponentnearthe transitionto chaos.As previously shown,fluctuationsproduceagap in theperiod-doublingbifurcation sequence.We showthat this implies ascalingbehaviorfor thechaotic thresholdanddeterminethe associatedcritical exponent.By consideringfluctuationsas a disorderingfield on thedeterministicdynamics,we obtain scalingrelationsbetweenvariouscritical exponentsrelatingtheeffect of noiseon theLyapunovcharacteristicexponent.A rule is developedto explain theeffectsof additivenoiseat fixed parametervalue from thedeterministicdynamicsat nearbyparametervalues.

* Presentaddress:Centerfor NonlinearStudies,MS B 258, Los AlamosLaboratories,Los Alamos,New Mexico 87545,U.SA.

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J.P. Crutchfield eta!., Fluctuations and simple chaotic dynamics 47

1. Introduction

Chaotic dynamical systems provide a significant new addition to the conventionaldynamicalrepertoireof equilibrium and periodic oscillation. Central to their usefulnessin describingobservedrandombehavioris the issueof their stability andthe observabilityof their bifurcation sequencesin thepresenceof noise sources.In particular,given the fact that fluctuationsare an importantaspectofmany-bodysystems,one would like to understandthe role of fluctuationsvis a vis the chaoticbehaviorgeneratedby deterministicnonlineardynamics.

Recentlywehaveshown [1] that for a certainclassof processesthe effectof externalfluctuationsonthe onset and propertiesof chaoscan be describedin fairly simple fashion. For systemsdisplayingperiod-doublingbifurcationswefound that the presenceof noiseleadsto both a gapin the bifurcationsequenceand a renormalizationof the chaotic threshold. Furthermore,we noticed that for fixedparametervalues the addition of noise to a nonlineardeterministicequation producesadditionalbifurcations,twhich can beobservedin measurablepropertiessuchas powerspectra.Theseeffectsareof importanceto current attemptsat understandingturbulentbehaviorof fluids andsolidsin terms ofchaotic,deterministicmodels.

In this paperwediscussin detail the effectsof fluctuationson the cascadebifurcation to chaos.Bythe cascadebifurcation we shall meannot only the infinite sequenceof subharmonicbifurcations,ateachstageof which the period of a limit cycle is doubted,but alsothe symmetricbifurcationsequenceabovethechaoticthreshold,in which n bandsof achaoticattractormergepairwiseto form an n/2-bandattractor.Ample illustration of the cascadebifurcation follows in the nextsection.

Previouswork discussedthe effect of external fluctuationson the cascadebifurcation found in adriven nonlinearoscillator [1]. That study was motivated in part by the fact that in condensedmattersystemsthermal fluctuationsplay an importantrole which had to be incorporatedinto the nonlinearequations leading to solid-state turbulence [2]. More generally, though, the cascadebifurcationsequenceandits alterationin thepresenceof fluctuationsis of interest in systemsthat rangefrom fluidflows [3,41to noisephenomenain solid-statesystems[5]. Moreover,this bifurcationsequenceis foundin numericalstudiesof a wide rangeof mathematicalmodels,including nonlinearordinary [6,7] andpartial [8] differential equations.In the experimentalobservationof bifurcation sequencesin Bénardflow [3] and sphericalCouetteflow [4] one observesonly a finite number(<4) of bifurcationsin thecascadesequence,whereasthe scaling theory developedby Feigenbaum[9] andCollet andEckmann[10] for the period-doublinghalf of the cascaderequiresthat the dynamicsundergoan infinite numberof bifurcationsbeforethetransitionto chaos.Indeed,it is in just thislimit of infinite bifurcation that thescaling theory becomesexact. As previously suggested[1], this discrepancycan be explainedby theinteractionof externalfluctuationsandthe deterministicsequenceof bifurcations.

From a study of the geometryof the attractorsfound in the driven anharmonicoscillator [1], wefoundthat increasednoiselevelscould inducea transitionto chaoticbehavior.Furthermore,ratherthandestabilizingor erradicatingchaoticmotions in the phasespace,noiseenhancedthe chaoticbehavior,while destroyingperiodic orbits.That is, the local instabilities responsiblefor the deterministicchaoticbehavioractually increasedthe observability of chaosin the presenceof fluctuations. Using Feigen-baum’sscaling theory andthe existenceof the bifurcationgap,we deriveda scalingrelationshipfor thenoise dependenceof observablebifurcations in a cascade.These featureswere also found in aone-dimensionalmap typical of thosefor which the universalscalingtheory wasdeveloped[ii.

Thepresentpaperreportsin moredetail theresultsof our studyof one-dimensionalmaps,that is, of

t weemploy abroaderdefinition of bifurcation thanis typical: anobservable,qualitativechangein a system’sbehaviorasa control parameterisslowly varied.

48 J.P. Crutchfield et al., Fluctuations andsimple chaotic dynamics

nonlinear transformationsof the unit interval onto itself. The main results of our paper can besummarizedas follows. In the absenceof noise, we verify numerically the scaling predictionsofHubermanandRudnickfor the behaviorof the Lyapunov characteristicexponentA anddiscussseveralotherfeaturesof A found in the cascadebifurcation. The Lyapunov characteristicexponentcan bethoughtof as a disorderparameterfor chaos[11]. We show that the existenceof a bifurcation gapimplies ascalingbehaviorfor thechaoticthresholdanddeterminetheassociatedexponent.By realizingthat fluctuationsact as a disorderingfield on the deterministicdynamics,we obtain a scalingrelationbetweenvariouscritical exponentsrelatingthe effect of noiseon A, anddevelopthe notion of a noisesusceptibility.We also show that even in the periodic regimethereis a non-trivial effect of externalfluctuationson the period-doublingbifurcations.This is reflectedin the fact thatnot only higher periodsbecomeobliterated by increasingnoise, but the bifurcation points themselvesbecome blurred. Inparticular,at the pointsof bifurcation, A no longer vanishesas it doesin the deterministiclimit.

In section2 we review the dynamicsof mapsof the interval in the absenceof noise.We presentatypical mapwhich displays the cascadebifurcation and analyzeit in detail. We summarizethe scalingtheory of bifurcationsanddiscussthe Lyapunov characteristicexponentas a measureof the stability ofthe asymptoticbehavior.Section 3 considersthe effectsof noiseon the cascadebifurcation sequenceandthe onsetof chaos.A rule is developedthat allows the effectof noiseata fixed parametervaluetobepredictedfrom the knowledgeof the purely deterministicdynamicsat adjacentparametervalues.Insection4 wediscussthe role of noiseas a disorderingfield andthe scalingbehaviorof the noisecriticalexponents.Section 5 containsa summaryof the scalingideasand discussesthe applicability of theseideas to physical systems.We then mention other questionsrelated to the interaction of chaoticdynamics,external fluctuations,and observationalnoise.A set of appendicesdiscussesdetails of theeffect of fluctuationson adriven anharmonicoscillator,the scalingof thecharacteristicexponentwithinthe period-doublingregime, andthe noiseequivalencerule of section3.

2. Dynamicsin the absenceof fluctuations

Dynamicalsystemstheory [121describesthetime evolutionof asystemas a trajectory,or an orbit, ina phasespaceof the system’spossiblestates.Typically, the physically interestingbehaviorof a systemisthat which is observedafterinitial transientshavediedaway.The set of stateswhich an orbit eventuallyvisits is called the system’sattractor.The studyof dynamicalsystemsconsidersnot only the structureofattractorsbut alsothequalitativechange,or bifurcation, from onetypeto anotherassomeparameterissmoothlyvaried.

Sincethe first physically-motivatedstudyof chaoticdynamicsby Lorenz [13], one-dimensional(1D)mapshaveplayeda fundamentalrole in the field’s developmentdespitetheir apparentsimplicity. The1D map obtainedfrom a system of ordinary differential equationscapturesthe essentialgeometryunderlying the chaotic dynamics.Although such a reduction of dimension (from three to one, inLorenz’s case)cannotbe uniformly appliedto all dynamicalsystems,for manyproblemsthe techniqueprovides more than sufficient heuristic insight into the processesresponsiblefor chaotic behavior.Specifically,by identifying all the pointswhich asymptoticallymerge, that is, all the pointson the samestablemanifold, it is possibleto summarizemanyof the propertiesof a simplechaoticattractorin threedimensionsby a ID map.

For dissipativesystemswith a chaoticattractorthat appearslocally two-dimensional,a crosssectionthroughtheattractorintersectsit in somecurve. Onecan thenconsiderthedynamicsas amapfrom thiscrosssectiononto itself; this mapis called the Poincarémap. By parametrizingpointson the curveof

J.P. Cnachfie!deta!., Fluctuations andsimple chaotic dynamics 49

intersection (from 0 to 1) and collecting a set of successive points {x1, x2, x3,.. .} as an orbit passesthrough the section, the dynamics of the attractor can be summarizedin a one-dimensionalmapof theform

x,,~1 = f(x~), n = 1, 2,3, - . . , (2-1)

wherex~is the nth crossingof theorbit throughthe sectionandwheref is a nonlinear function on theunit interval.In thissense,discretetime mapssummarizethedynamicsunderlyingchaoticbehaviorfoundin higherdimensionalsystems.Fromtheir simplicity, 1Dmapshavedevelopedasprototypicalmodelsin thestudyof chaoticdynamics[14].

For dissipative dynamicalsystems,such as discretemappingsand ordinary and partial differentialequations,that exhibit cascadingbifurcations,the dynamicscan be describedin practiceby a 1D mapwith asingle smoothmaximum.An exampleof suchmapsis providedby thelogistic equation,which isdefinedby

Xn+irxn(1Xn), O<Xn<1, (2-2)

and where the bifurcation parameter r (0< r <4) determines the height of the quadratic functionf(x) = x(1 — x). As a graphic example of the complexity present in this class of maps,the bifurcationdiagram of fig. 1 presentsthe changein the attractorof eq. (2-2) as a function of the bifurcationparameterr in the regime[3,4].

At afixed parametervaluein the bifurcation diagram,a periodic orbit consistsof a countableset ofpoints, while a chaotic attractorfills out densebandswithin the unit interval. Figure 2 shows theprobability densityfor the two bandsat r = 3.59687.The dominantbifurcation sequenceseenin fig. 1 isthe single2~cascade,through which theattractorfirst becomeschaoticandeventuallyfills the intervalvia the pairwise mergingof bands.The period-doublingand band mergingaccumulatesat a value

P-2 P-4 P-B P-7 P-B P-4I I

{x~} ~

03.0 r r~p-~ P-5 P-3 4.0

Fig. I. The attractorversusbifurcation parameterr for the logistic map, eq. (2-2), x~+1= rx,,(1 — x~).700 iterationsplotted after an initial 500iterationsfor eachincrementin thebifurcation parameter.The parameterwas incremented1000times in the interval [3,4]. For thesakeof clarityandresolution,only thebifurcation diagramfor r in [3,4] is shown.For r in [0, 1], x~= 0 is thestablebehavior;andfor r in [1,3], onehasastablefixed pointdescribedby x = (r — 1)/r.

50 IP. Crutchfield et al., Fluctuations and simple chaotic dynamics

4.0

r=3.59687

Fig. 2. Normalizedprobability density P(x) of thetwo band attractorat r = 3.59687 shown on a logarithmic scale.P(x) is a histogram of i07

iterationsof eq. (2-2) partitionedinto 1000 bins.The peaksin P(x)aresuccessiveimagesof themaximum,or critical point, x~.Note that eachbandis a mirror imageof theother.

r~= 3.569945672...after an infinite numberof bifurcations. In the chaoticregimeabove r~,onefindssmallwindowsof higherperiod cascadeswith periodsq x 2~,with q an integerandwheren denotesthedegreeof the period-doublingof a fundamentalperiodic orbit of period q. Within eachsuchwindow,onealsofinds the associatedreversebifurcation [7] of q x 2”~’bandsmerginginto q X 2” bands.In whatfollows we will call q the periodicity of the cascade;q = 1 for theprimarycascadedescribedabove[15].

To describein moredetail the structureapparentin fig. 1 we now focuson the successiveimagesofthemap’smaximum,calledthemap’scritical pointx~,wherethe slopevanishes.Oneof themorestrikingfeaturesof thebifurcationdiagramabover~is theveil-like structurehighlightedby dark lineswhich varysmoothlywith theparameter.As theattractorsin thechaoticregimeconsistof densesubsetsof theintervalratherthandiscretepoints,oneneedsto considerthe actionof the mapon aprobabilitydistribution.Thedarklinesin thediagramcorrespond,then,to successiveimagesof thecritical pointandindicateregionsofhigh probability density.Theseare seenin fig. 2 as spikesin the probability density.To describetheirdependenceon the bifurcationparameterr, it is usefulto write

x~÷1= F(r, x~), (2-3)

with F(r, x) = rx(1 — x) in our example. Then the mth image of th~critical point, x~= 0.5, is apolynomialin r, F

m(r, 0.5),whereFm(r, x) = F(F”’(r, x)). For example,abover~,the first imageof x~definesthe upperboundon {x~};it is the straight line

F(r, x~)= r/4, (2-4)

seenin fig. 1. Similarly, the secondimagedefinesthelower boundon {x~}which is given by

F2(r, x~)= ~- (i — ~). (2-5)

Furtheriteratesof the critical point must lie betweenthesetwo. For example,the third iterate

J.P. Crutchfieldet a!., Fluctuations and simple chaotic dynamics 51

F3(r, x~)= ~ (i — ~) (i — ~- (i — (2-6)

is a lower boundon the upperbandin the two-bandregion.Similarly, the fourth iterateF4(r, x~)is anupperboundon the lower bandin the two bandregion.Thepair of bandsmergesinto oneband at aparametervaluermergedeterminedby the intersectionof F3(r, x~)andF4(r, xe), that is, where

F3(r, x~)= F4(r, x~)= F(r, F3(r, xe))

or (2-7)

1 = r(1—F3(r,x~))

whichgives rmerge= 3.67857351-... It is now apparentthat eachof thedark lines in the chaotic regionof the bifurcation diagramcorrespondsto oneof the imagesof the critical point Xc.

The appearanceof stableperiodic orbits within the chaotic regimeis signalledby the qth iterateofthecritical pointmappingonto itself; that is,

Fm~(rx~)= Fm(r, xe), m = 1, 2,3 (2-8)

As can be seen in the diagram,the labeledq-periodic regimescoincidewith the crossingof the qthiterateof the critical point F~(r,x~)through x~= 0.5. In fact, the period q orbit becomesstablejustbeforethis pointwhich is thepoint of superstability.The numberof windowsof aparticularperiodicityincreaseswith q and eachof thesewindowsis generallyof differentwidth. Furthermore,the width ofthesewindowsdecreasesrapidly with increasingq [7].

In the chaoticregime,onecan seethe unstableremnantsof the orbits of period 2” whichwerestablebelow r,~.Indeed,oncean orbit comesinto existenceat a given parametervalue it doesnot vanishathigherparametervalues,althoughits stability may change.Theseremnantsareapparentas particularlylow valuesin the probability densitywithin the chaoticbandsandemanatefrom thepointsat whichthebandsjoin. Thewhite streakscorrespondingto theseunstableorbits areseenmost readilywhenviewingfig. 1 from the side,looking parallelto the r-axis in the direction of decreasingr.

Despitethe existenceof suchdetailedstructure,the bifurcation diagram exhibitsa high degreeofself-similarity. By self-similarity we meanthe propertyof objectswhosestructure,as observedon onelengthscale,is repeatedon successivelysmallerscales[16].To describethis, we denotethe valueof r ata bifurcation by rmn, where, if m <n, a periodic orbit bifurcatesfrom period m to period n and, ifm > n, m bandsmergeinto n bands.If we considerthe bifurcation diagramfor r in [1,4] as the firstscale,wherethe attractorbeginsjust above r = 1 as a period 1 orbit and endsat r = 4.0 as a singlechaoticband,then this structureis repeatedtwiceon a reducedscalewithin the parametersubintervaj_[r

1_2,r2_i] = [3, 3.67857.. .], four timesin the still smallerregime[r2..4,r42] = [3.44944..., 3.59257...]andso on. In this manner,the period 6 regimecan be thoughtof as two copiesof the period 3 regime.Thus one need only describein detail the periodic regimesin [r21, 4] = [3.67857... , 4] in order tounderstandthe periodic regimesin [re,r21] = [3.56995..., 3.67857.. .]. Within this scheme,the period7 window just below r = 4.0 gives rise to a period 14 counterpartjust below r21. Suchself-similarity isalsofoundwithin eachwindow of higher periodicity.

Feigenbaum [9] has developed a scaling theory for the non-chaoticperiod-doublingside of thecascade bifurcation in 1D maps which exactly describesthis self-similarity in the limit of highly

52 J.P. (‘rutchfleld et a!., Fluctuations and simple chaotic dynamics

bifurcatedattractors.This theory predictsthatthe parametersr,, at the bifurcation from a period2” to aperiod 2” ~‘ orbit scaleaccordingto

(re— r,,)— es” (2-9)

where 8 = 4.69920.. . dependsonly on the (quadratic) nature of the maximum of the map. It alsopredictsthat the width of bifurcation “forks” in the periodic regimewhich stradlex~decreasesby aspatial rescalingconstant a = 2.502907876....Lorenz [7], in turn, hasshown that the bifurcationparametersfor the bandjoinings in the chaotic regimealso scalein this mannerandthat the width ofthe bandcontainingx~scalesby a atband-mergingbifurcations.GrossmanandThomaewereapparentlythe first to measurethe scalingfactor8 for both the period-doublingandband-mergingbifurcations[15].Furthermore,thebifurcationparametersdescribingthecascadesin all of thewindowsof higherperiodicityexhibit this scalingbehavior.

In the studyof chaoticdynamicsit is often usefulto havesomemeasureof thedegreeof randomnessgeneratedby the deterministicequations.Severalquantitieshavebeendevelopedfor this, but we shallonly consider those related to orbital stability. Orbital stability for a given orbit dependson thebehaviorof its neighboringorbits. If points nearan orbit convergetowardit, then it is stableto smallperturbations and is said to belocally stable.An orbit will be attracting,that is asymptoticallystable,ifon the a~veragealongthe orbit it is locally stable.If neighboringpointsdivergeaway from an orbit, thecorrespc~ndingbehaviorwill be sensitiveto smallperturbations.In this case,the instability will amplifythe perturbationsandtheorbit will be locally unstable,evenif the orbit initially convergedtowardsomeattractor.

In the caseof 1D maps,thesestability criteriaaremeasureddirectlyby theslopeof the mapatpointsvisited by an orbit. In particular,if the slopeat a point is lessthanone, nearbypointswill be broughtcloserto it at the next iterationof the map.Similarly, if the slopeat apoint is greaterthanone,nearbypoints will be spread apart under iteration of the map. An asymptoticallystableorbit, then,requiresthegeometric average of theseslopesalongtheorbit to belessthanone.Whenthisaverageisgreaterthanonetheorbit is unstableand,consequently,initially smalldeviationsfrom theorbitwill increaseunderiterationof the map.

This sensitivity to small deviations has important consequences for the physical behavior associatedwith chaoticdynamicalsystems. For if these deviations are due to some initial uncertainty in specifyingor measuringa state, then this uncertaintywill grow (exponentially, at first) until one can no longerpredict the state of the system within the attractor. The informationaboutthe initial stateof the systemis lost in afinite amountof time andsothe systemis effectively unpredictable[18,19]. This sensitivitytosmallerrorscan beconsideredoneof thedefining featuresof chaos[20].This in turn leadsto a physicalinterpretationof the degreeof randomness,given by the averagelocal stability, as the rate at whichinformation about states is lost [18,19].

The measureof averagelocal stability, the LyapunovcharacteristicexponentA, can be expressedintwo different, but related,ways [21].The first is the information-theoretic entropy, given in the caseof1D maps [19] by

A(r) = J P(x)lnIf’(r, X~)Idx (2-10)

J.P. Crutchfield et a!., Fluctuations and simple chaotic dynamics 53

where P(x) is the asymptotic probability distribution of an orbit at a given parametervalue, such asshown in fig. 2, and!’ is the slope of the map. If oneassumesergodicityof the orbit within the attractor,thereis an alternativeform for computingA(r). It is given by

A(r) = In If’(r, x~)l. (2-11)

With this definition the measureof averagelocal stability is such that if(i) A(r)<0, the orbit is stable and periodic;(ii) A(r) = 0, the orbit is neutrally stable;

and (iii) A(r) >0, the orbit is locally unstableandchaotic.Many of the featuresfound in the bifurcation diagram of fig. 1 are reflected in the curve of the

characteristicexponentas a functionof r. Thiscurve is shownin fig. 3 for theparameterregimeof fig. 1.For r < r~,A (r) � 0, indicating the existenceof periodic orbits only. In this regime,the period-doubling“pitchfork” bifurcationsoccurwhereA vanishes.That is, an orbit must first passthrough a neutrallystableattractorbefore it can takeon a qualitatively different structure.Betweenthesebifurcations,Aapproaches—oc as the critical point x~becomesa point on the periodic orbit. The orbit is said to besuperstable.The logarithmic divergenceof A at this valueof the parameteris easily inferredfrom eq.(2-10)or (2-11)becausethe slopeat the critical point vanishes(seeappendixB). Therelaxationof initialtransientsonto the periodicorbit differs on eithersideof thesesuperstablebifurcations.For r less thanthe bifurcation value TsuperstabIe, the approachof an initial transientis (eventually) from only oneside;while for r> rsu~rstab1e,a transientorbit alternatesfrom onesideof pointson the periodic orbit to theother.For example,the period 1 orbit becomessuperstableat r = 2. For r <2, any initial conditionwillapproachfrom the left of x~,even though it may havestartedfrom the right half of the interval. Forr >2, initial transientsalternatesides as they approachthe periodic orbit. The period 2 orbit whichbecomessuperstableat r = 3.236068...consistsof two points. The transientsin this casefor r <

Tsuperstabie eventuallyapproachonly from the lower side of each point, while for r> rsui,~ystabIetheyapproachthe points of the attractor from each side. The changein stability associatedwith thesuperstablebifurcation is clear in the A(r) curve, despitethe fact that it is not at all visible in thebifurcationdiagramof fig. 1.

:2.03~~

Fig. 3. Characteristicexponentversusbifurcation paranset~rfor the logisticequationcalculatedusingeq. (2.11)for 30000 iterationsateachof 7000incrementsof r in [3,4].

54 J.P. Crutchfie!d eta!., Fluctuations and simple chaotic dynamics

Finally at r = r~,A = 0 and one sees a bifurcation to chaoticbehavior indicatedby positive A forr> r~.Hubermanand Rudnick [22] have recently shown that the envelopeof A near r~displaysuniversalbehaviorreminiscentof an orderparameternearthe critical point of aphasetransition.Thatis, onecan write

A(r) = Aq(r — r~)~ (2-12)

with r = In(2)/ln(8)= 0.4498069...and Aq a constant.Cascadesof higher periodicities in the chaoticregimeareindicatedby windowsof negativeA in which A goesthrougha period-doublingsequenceofpitchfork andsuperstablebifurcations.Above the accumulationpoint in eachwindow the envelopeofpositiveA scalesas in eq. (2-12) exceptthat the constantAq dependson the periodicityq andthe widthof thewindow. Fromnumericalstudiesof the characteristicexponentusingeq. (2-11), wehaveverifiedthis scaling behaviorfor various periodicities. For example, for q = 1, 3 and 5, r = 0.45+ 0.01, andA1 = 0.84, A3 = 1.7 andA5 = 1.6.Figure4 showsA(r) andeq. (2-12)graphedwith the valuesfor q = 1 and3.

An estimateof Aq can be basedon the observation that at band joinings A is inversely proportionalto the numberof bands.Specifically, at a bifurcation from q 2” to q - 2”’ bands,the characteristicexponentis given by

~ r~)~ ‘2-13‘‘ q2” qk’ ‘

that is,

(214)

where the approximation is less than a few percent and k is the constantof proportionality of (2-9)

which depends on the periodicity q [23].

I.0

r r~3.885

Fig.4. CharacteristicexponentcurveAfr) from 11g. 3 on anexpandedscalefrom r~to r = 3.885.Thetwo smoothcurvesshowthefit of eq. (2-12) to theenvelopeof theq 2” cascades.For q = I andq = 3, wefind A, = 1.01 andA3 1.7 in fitting eq. (2-12).Thecurvefor theq = 3 chaoticregimehasbeenextendedbeyondits rangeof validity to makeit easierto seeits fit to theenvelopeof positivecharacteristicexponent.

J.P. Crutchfie!d et a!., Fluctuations andsimple chaoticdynamics 55

When one looks for the featurein A (r) correspondingto bandsmerging the self-similarity againbecomes apparent in the A(r) curve of fig. 3. There is a peculiar upturn of A(r) toward ln(2) near r = 4.0[24].This upturn also displays a critical behavior given by [10]

A(r) -= 1 — b(4.0— r)~2 (2-15)

whereb is a constant.In this regime, thereis a conspiciousabsenceof negativedips in A(r) indicatingthe existenceof little observable periodic behavior. These features also occur for values of r <4.0 atwhich bandsmerge.Severalof the corresponding“steps” in A(r) can be seenin fig. 3. Oneof them isshown magnified in fig. 5 near the merging of two bandsinto one. Using eqs. (2-13) and (2-15), weobtain the following expression for A(r) near band joinings

A(r) = Aq(r — r~)~(1 — bq(Tband — r)112) (2-16)

where bq dependson the periodicity q. Thus,as onewould expect,the entire A(r) curveexhibitsthesame type of self-similarity as the bifurcation diagram. A(r) in eachof theseself-similar subintervalsisvery near one-half of that in the preceding subinterval. Again, knowledge of the structureof the A(r)curve in [r

2..1,4.0] is sufficient to (qualitatively)determinethe entirecurvefor [re,4.0].Although A (r) is continuousin the chaotic regime, it does not convergeto a limit curve with

increasingresolutionin r. It is a curve of infinite length that admits of no closed form representation.Some of the featuresassociatedwith this propertycan be describedby different typesof self-similarity.Furthermore,theseself-similar featurescan be describedby a “fractal” dimension[161.

To emphasizetheunusualnatureof the A (r) curve, we shallmentionthreeoccurrencesof self-similarstructure. The first is the global self-similarity found in the bifurcationdiagram.The featuresbetweenr2_1 and r 4.0 are repeatedon smallerscales(reducedby a factor of 8) as one approachesr~fromabove. This self-similarity has already been discussed above. Another property of the A(r) curve in thechaotic regime is that at every degreeof resolution in r, thereare windows of periodic behaviorcorrespondingto the negativedips in A(r). The width of the windowsrapidly decreaseswith increasingperiodicity q. The numberof windowsobservedat a given resolution in r appears,from our numerical

0.4

X(r)

02~__3.65 r r21 3.7

Fig. 5. A(r) neartwo bandsmerginginto oneat r2_1 = 3.67857.... The kink in A(r) at r2~as seen in thefigure is characteristicof bandsmerging.Notethat A(r2_,)~-ln(2)/2 —0.35. This pictureis takenfrom thedataof fIg. 3.

56 fl’. Crutchfield eta!., Fluctuations andsimple chaotic dynamics

work, to be roughly independent of the resolution at which A(r) is studied. This is the secondself-similar featureof A(r): a local self-similarity. It dependson the parameter r and gives a measure ofthe densityof periodicwindowsin the chaoticregime.

Thesetwo features,andthe self-similarcharacterthey lendto A(r), canbe summarizedby thefractaldimensionof the A(r) curveitself. We havemeasuredthefractal dimensionof A(r) for r in [re,41 usingastandardalgorithm discussedby Mandeibrot[16]. The fractal dimensionof a curvedescribeshow thecurve’s length increaseswhen measuredon smaller scales.To apply this to the curve of fig. 3, wemeasurethe lengthof A(r) in units of s ~r, wheres is somemultiple of the increment~r in parameterused in making fig. 3. The fractal dimension/3 is obtainedthen by varying the size of the measuringunit. The number N of measuringunits of size s ~r is given by

N = As~, (2-17)

whereA is someconstantdependenton t~r.The resultsof theselengthmeasurementsareshownon alog—log plot in fig. 6. Wehave taken s = 2”, n 0, 1,..., 11, andfound/3 = 1.69 and A = 4.63 X iO~for rin [re,4]. The fact that b is not 1, as would be the case for a simple smooth curve, indicates that as themeasuringunit is decreasedby 2 (say), morethantwice as manyunits arerequiredto coverA(r). Thus,as the resolution is increasedmore featuresin A(r) becomeapparent,so that in the limit the length,1 = Ns~r, of A(r) is infinite.

The last self-similar feature we shall mention is the fractal dimensionof the attractorsthemselves.Periodicorbits and chaoticattractorshavetrivial fractal dimensions0 and 1, respectively.The formerconsistof acountablenumberof discretepointsandthe latterfill out densebandsin the interval.At thetransitionsto chaos,however, whereA (r) vanishes,the attractorshave a self-similar structure:theuncountablenumberof pointson theseorbits aredistributedby factorsof Feigenbaum’sspatialscalingconstanta. The fractal dimensionof theseattractorshasrecently beencalculatedto be dtransition=

0.538...[25].These self-similar featuresare substantiallymodified in the presenceof fluctuations. The fractal

dimensionof each,though, still gives a qualitativemeasureof the level at which fluctuationstruncatethe self-similar structure.We shallreturn to this point at the endof the following section.

0.0 Ins 13.0Fig.6. Log—log plotof the lengthof theA(r) curve,for r in [re,4] versusthescaleof measurements.N is thenumberof measuringunitsof length snecessarytocoverA (r),wherei~ris theincrementin C usedto computeA (r) in fig. 3. Accordingto eq.(2-17), theslopeoftheline is thefractaldimension~.

For r in thechaotic regime [re,4] we havefound thefractal dimension8 = 1.69.

fl’. Crutchfield et al., Fluctuations and simple chaotic dynamics 57

As a concludingremark, we should point out that self-similarity also leads to scaling predictionsrelatingto the behaviorof the powerspectraassociatedwith deterministicnoise[26]. Onceagain,onecan associateuniversalexponentswith both the growth of deterministicnoise as a function of thecontrol parameter and the width of the bands past the bifurcation cascade.

This endsour review of the dynamicsof 1D mapsin the absenceof fluctuations.In thissection,wehaveemphasizedthe self-similarity of the dynamicsas a function of the bifurcationparameterr. In thefollowing sectionsweshall discusshow this self-similarityallows for ascalingdescriptionof the effectoffluctuationson the cascadebifurcation. In particular, we will study the effect of noise on the fractalattractorat r~and will find that noise can be describedby a scaling theory, suchas foundin the theoryof critical phenomena. This noise scaling is also of interest in the study of the effect of noiseon thefractal structure of chaoticattractorsin higherdimension,evenawayfrom period doubling bifurcations.

3. Dynamicsin the presence of fluctuations

The nature of the fluctuations introduced into a dynamical model depends on the coupling betweenthe physical process to be describedand the sourceof randomperturbations.In describinga physicalsystemspecifiedby a set of differentialequations,dynamicalsystemstheory considersthe actionof aflow on a suitablephasespaceof states.Fromthis perspectivefluctuationscanenterin two ways: first,as an “external” stochasticforce that perturbsthe phasespacetrajectory; and second,as a randomperturbationof the parametersspecifying the flow itself. We shall call the first additive noiseand thesecond parametric noise. The analog of additivenoisefor 1D mapsis of the following form,

x~±1=f(r, x~)+p~ (3-1)

where p~represents a random deviation from the deterministic orbit. For parametric noise, thefluctuations perturb the form of the nonlinear function or the parameters in the map. For example, if wewrite the deterministic equation as

= rf(x~) (3-2)

with r as the bifurcationparameter,then parametricfluctuationswould be of the form

= (r+ q~)f(x~) (33)

or

= rf(x0)+ q~f(x~) (3-4)

with q~representingthe fluctuationsin the parameterr. In the caseof 1D maps, the two physicallydistinct typesof fluctuationreduceto basically the sameform, namelythat of eq. (3-1), exceptthat eq.(3-4) hasdifferent statisticsfor the “external” stochasticforce: q~f(x~) ratherthan just p,,. This willintroduce higher-order correlations, but for small fluctuationsthe systemsof eqs. (3-1) and (3-4) willexhibit the samebehavior. In what follows we shall discuss the effect of changesin the level offluctuations on the behavior of eq. (3-1) with f(r, x) = ix(1 — x). We shall take p~ as a Gaussian or

58 J.P. (‘rutchfield etal.. Fluctuations and simple chaotic dynamics

~x~}

03.0 r 4.0

Fig. 7. Bifurcation diagram in thepresenceof zero-meanGaussianfluctuationswith standarddeviation,or noise level, ci = iO~.Computationaldetailsarethesameasin fig. 1, exceptthat eq. (3-1) was used.

uniform random variable with standarddeviationu andzeromean.Wehavefound that the resultsdonot depend significantly on the choice of distribution. We will refer to o- as the noise level of thefluctuations.

A heuristicdescriptionof fluctuations,asimposingaminimumscaleof deterministicresolution,leadsone to properly anticipate the bifurcation diagram in the presenceof noise. Figure 7 illustrates theeffects of noise on the 1D map, with p,, a Gaussian random variable of standard deviation ~- = iO~.Ascan be seen in comparing figures 1 and 7, the fluctuations truncate the detailed structure by smearingthe sharpfeaturesof the probability distribution of the attractorsseenin fig. 1. An exampleof thiseffect on two bandsis shownin fig. 8 for r = 3.7 ando~= iO~.This shouldbe comparedto fig. 2. Theperiodic regime, where the orbits are slightly broadened, is easily distinguished from the chaotic regime,wherethedistribution of pointswithin the bandsappearsmoreuniform thanin fig. 1. Whenfluctuationsof sufficient amplitude are added, periodic orbits broaden into bands similar to chaotic attractors. Asonewould expect,the reverseprocessof chaoticattractorsturninginto periodicorbits doesnot happen.

0.0‘Th L,Fig.8. Logarithmicplot of the(unnonualized)probabilitydensityP(x)of thetwo bandattractorat r = 3.59687in thepresenceof ci = iO~noiseasin fig. 7. Comparethis with fig. 2 in which thereis no addednoise.P(x) is a histogramof iO~iterationsof eq. (3-1) with p~a uniformly distributedrandomvariableof standarddeviationci = iO~.The pointshavebeenpartitionedinto 10’ bins.

J.P. Crutchfieldeta!., Fluctuations and simple chaotic dynamics 59

In other words, fluctuationsgenerally increasethe degreeof randomnessor chaosin the dynamics,while destroyingthe periodic windows encounteredin the chaotic regime. This enhanced“obser-vability” of the chaotic behaviorin the presenceof fluctuationsoriginatesin the local instabilitiesunderlying the deterministic chaos. Thus fluctuations affect the local stability propertiesof theattractors, while leaving their global stability relatively unchanged.

Some of the structureobservedin the deterministiclimit (o- = 0) is still visible abover~when noise isadded. In particular, it is evident from fig. 7 that in the chaotic regime a period three orbit is stillpresent, although,windows of higher periodicity are not. The cascadesof higher periodicity becomeunobservable at noise levels whose intensities are proportional to the parameter width of the window.Also, as shown in figures 7 and 8, the images of the critical point still appear as regions of higherprobability, but with fewer of the higher iteratesencounteredin the deterministiclimit. Furthermore,for the noiselevel of fig. 7, a period of four is the highest that remains in the primary 2” cascade,whileall higher periods are washed out.

To summarizetheeffectsof noisefor a given cascade,we will define~ as the noise level that resultsin a maximum observable period p. This quantity determines a lower boundon thoseperiodswhichhave been washed out. The effect on the primary 2” cascadeis illustratedby a diagram,shownin fig. 9,displaying a gap in the observable periods which increases with noiselevel. Thisgaprepresentsthe setof attractors(both periodicorbits andchaoticbands)of periodsgreaterthanp which areinaccessibleataparticular noise level o~,. Figure 9 showsthe qualitativedependenceof the gap on the noise level,wheretheverticalaxis denotestheperiod p = 2” of periodic orbitsor thenumberof bandsp = 2” of thechaotic attractor. The deterministic limit a- = 0 correspondsto the full cascadebifurcation. Withincreasing noise level, though, a symmetric gap in the bifurcation sequenceappears, renderingunobservable successively more periodic and chaotic bifurcations. The gap in fig. 9 symbolizes a set offorbidden attractors whose periods cannot be observed at a given noise level. Both the largest

Chaotic

2 ~//hllh/llhI//I//I/I/III/I/III//I/II/I///IAP ~g ii~iiiii~

4 ~- iiiiiiiii~8 ~.~~#i/ii1/IIihI/IIIIIIIIJI//I/II/II//I/I/II/I/III1~”I1I.II/jiiiii/,j1f/jii/i///jfjii4

2 Periodic

1 I I I0 0.01

Fig. 9. The setof observableattractorsfor thecascadebifurcation asa functionof (normalized)noiselevel. Theverticalaxisdenotestheperiodp ofanattractorgiven(I), in thecaseof adriven oscillator[1],by theratio of theresponseperiodto theperiodof thedriving force,or (ii), in general,bytheperiod(numberof bands)of theperiodic(chaotic)attractor.Thenoiselevelci alongthehorizontalaxiscorrespondsto thestandarddeviationofthermal-likefluctuationsin which thesystemis immersed.Theshadedarearepresentstheset,or “gap”, of unobservableattractorsobliteratedby agiven levelof fluctuations.

64) .J.P. Crutchfield et a!., Fluctuations and simple chaotic dynamics

observableperiodandmaximumnumberof observablebandsdecreasewith increasingnoiselevel a-. Asan exampleof the alterationof the cascadebifurcationwith noiselevel, considerstartingin fig. 9 from aperiod 4 orbit at a- 0. As the noiselevel is increasedfrom this point, the gap is finally reachedata- -~-~5 x i0~wherethe attractor appears almost like a 4 band chaotic attractor. The attractors observedat highernoiselevelsarethosealongtheupperboundaryof thegap,within the chaoticregime.That is,increasing noise levels strengthenthe band-like characterof the attractor, eventually inducing atransition to a chaotic attractor. This transition from a noisy periodic attractor to a chaotic onecorresponds to a vertical jump across the gap. The attractor reaches another transition point ata- -= 10~wherethefluctuationshavebecomestrongenoughto broaden the four chaotic bandsinto twobands.A similar transition takesplace, in principle, at a- -~2 x 102 where the 2 bandattractorwillbroaden into one band. Similarly, at a fixed noise level in fig. 9, one encounters a transition to a chaoticattractorat parametervalueswhich decreasewith increasingnoiselevel. Thus the existenceof thegapalsoillustratesthe fact that fluctuationscan induce a transitionto chaosat a lower threshold.

Since the exact values of a- and p for which a gap appears depends on the determination of when agiven period becomesunobservablewith increasingnoise, we havecalculatedthe Lyapunovcharac-teristicexponentcurveto provideaconsistentmeasureof theonsetof chaosasafunction of noiselevel.When noise is added,the distinction betweenperiodic orbits and chaoticattractorsis no longer asstraightforward as in the deterministic limit: all attractors eventually fill out intervals whose widthdepends on the noise level. Nevertheless,there is a qualitative difference between the stabilityproperties of the two types of noisy attractors, a difference which is reflectedin the behaviorof thecharacteristic exponent. In calculatingthe characteristicexponentin the presenceof noise,we usedeq.(2-il) with f’ the derivative of the deterministic mapevaluated at points alongan orbit {x0} takenfromiteratesof eq. (3-1).

Figure 10 shows our computation of A(r) at the samenoiselevel as the bifurcation diagram of fig. 7.When this is compared to the deterministic case, fig. 3, several changes are immediately apparent. First,the small parameter windows of periodic behavior in the chaotic regime disappear.Second, thehigher-order period-doubling bifurcationsmergeinto a singlesmoothcurvenearr~.Third, bifurcationsbetweenperiodic orbits,whereA = 0 in the absenceof noise,becomemorestablewhenfluctuationsareadded. Fourth, the first transition to positive A occurs earlier in the bifurcation sequence. We willcontinue to associate this latter feature with the onset of chaotic behavior. And finally, the value of A atparameter values that are chaotic in the deterministic limit is effected substantiallylessthan for the case

Fig. 10. Characteristicexponentversusbifurcation parameterr at thenoiselevel of fig. 7: ci = ~~_3• Detailsof thecalculationarethesameas in fig.3, exceptthat eq. (2-11)wasusedwith iteratesof eq.(3-1).

J.P. Crutchfie!det a!., Fluctuations and simple chaotic dynamics 61

of periodicbehavior.In mostchaoticregimestheaddition of noiseleavesA unchangedoversomerangeof noiselevel.

Figures 7 and 10 were constructedusing additivenoiseat fixed parametervalues.A study of thesefiguresmakesit clearthat it is difficult to anticipatetheeffect of addingnoiseunlessthebehaviorof thedeterministicsystem is known at adjacentparametervalues.Roughly speaking,adding fluctuationsaltersadeterministicorbit so that it wanders over points on the attractors at adjacent parameter values.This suggestsa simple model for the action of noise on the deterministicbehavior: the effect offluctuations is to average the structure of deterministic attractorsover some range of nearbyparameters.That is to say, there is an equivalence between a perturbationof an orbit and aperturbation of the map itself, in the sensethat at eachiterationthe effect of a perturbationof an orbitcan also be obtainedby a suitablechangein parameter.With this equivalence,all of thefeaturesof thebifurcationdiagram,fig. 7, andthe A(r) curve, fig. 10, can be understood,aswell asotherconsequencesof addingnoise.

To understandwhy this averagingof parametersis effective, consideran orbit {x0} of eq. (3-1),generatedby someparticularsequenceof fluctuations{p,,}. If justthe right sequenceof fluctuations{q~}occurs,an identicaltrajectorywill be producedby a parametricnoiseprocessof the form of eq. (3-3) or(3-4). This sequence{q~}can be found by equatingthe trajectoriesstepby step,that is,

= rf(x~)+p~= (r+q~)f(x~). (3-5)

For the casef(x)= x (1 — x) we obtain the following equationfor q~

~ (1—x,)~ (3-6)

That is, at each iteration the effectiveparametervalue is r’ = r + q~.For this particulariteration, thenoisysystembehavesjust like a purely deterministicsystemat parametervaluer’, exceptthat the pointx,, is not necessarilyon the attractorfor thisparameter.Therefore,the attractorof the noisysystemcanbe approximatedby an appropriately weighted averageof the deterministic attractorsat nearbyparametervalues.

As maybeseenfrom eq. (3-6),the statisticalpropertiesof {q,,} arequite different from thoseof {p~}.Sinceq~dependson x~,its statisticsmaybe as complicatedas thoseof x~.For example,supposethedeterministicattractoris a period 2 orbit andthe additive fluctuationsp,. are small. Becausethe twopointsof the orbit havedifferent values,eq. (3-6) saysthat the fluctuationsq~with odd n will on theaveragehave a different magnitude than those with n even. Thus, in this case, if the additivefluctuationsare ergodic, the parametricfluctuationsare not. In addition, {q~} neednot be Gaussian,evenif {p~} is.

To calculate the statistical propertiesof {q,,} exactly would require simulating the orbit {x~}.Nevertheless,very crude estimatesof a few moments of q~,together with a knowledge of thedeterministicbifurcations,providea good understandingof the effect of additivenoise.In what follows,wewill estimatethe first two momentsof the equivalentparametricnoiseq,,. We will then discusshowthis can beused to explain the observedeffectsof noiseon figures 1 and3.

We will now computethe averageof q~.Sincethe fluctuationsp,, arestatistically independentof x~we can write

62 J.P. C’rutchfield et a!., Fluctuations and simple chaotic dynamics

(q,~)= (p~J(x~(i—x~))) = (p~~(lI(x~(i— x,,))) = 0 (3-7)

where (. . .~‘ denotes time average and where we have assumed (p,,~ = 0. Thus the average of q~ is zero.Estimating the second moment is more difficult. Squaring eq. (3-6) and again making use of the fact

that p~and x~are uncorrelated, we obtain

~‘ 2(q~)= ~ (1/(x~(1— x~))~), (38)

which, unfortunately, depends on an unknown moment of x~. However, in many cases this moment canbe estimated. For example, if the deterministic orbit approaches a fixed point, and the additive noise issmall, the value of the fixed point can be used to obtain a good estimate of (q~). To do this for a limitcycle, the fluctuations about each point on the deterministic orbit must be considered separately, sincein this case the fluctuations q~are not ergodic.

A simpler, but less accurate, expression may be obtained by rewriting x,, (1 — x~)in terms of x,,+1,

x,, (1—x~) = (x~±1— Pn)/T. (39)

Squaring,substitutingthisinto eq. (3-8), and expanding the denominator,weobtain

2 ~‘

(q~)= ~(rp,,Jx~±1))~1—2p,,Jx,,±i +3(pJx~±1)—...) (3-10)

with p,, and x,÷1 correlated. If p. is small, however,the dominantfactordeterminingx,,÷1is x,,, and p,,

will be approximately uncorrelated with x~±1.With this assumption, we find

cr0 -~-ra-,~1/x2)”2 (3-11)

where 0q and o-~ are the standard deviations of q~and p

0, respectively. To estimate (1/x2), we define

t~x2 = x2 — (x2). If (1~x2)~ (x2), then (1/x2) can be expanded to give

O~q ~ (i — (~))h/2, (3-12a)

which is a good approximation for periodic orbits. For chaotic attractors there is no reason to assume(z~x2) ~ (x2) unless the probability density is sharply peaked or consists of bands. From numericalcalculations, however, the moments in eqs. (3-8), (3-11) and (3-12a) all agree to within less than a percentfor rin [3,4].

In practice, then, very coarse approximations are effective. For example, if we take r = 4.0 where thechaotic attractor fills the entire interval, approximate (x2)”2 by its median value of 0.5, and take z~x2= 0,we can obtain a crude approximation from eq. (3-12a) for the standard deviation of the equivalentparametric noise level 0~qfor a given additive noise level a-p. This estimate is given by

a-0 ~‘8cr,,. (312b)

With this equivalence, it is possible to construct a good approximation to the noisy bifurcationdiagram of fig. 7 from the deterministic bifurcation diagram of fig. 1. We have the following rule:

J.P. Crutchfield et a!., Fluctuations and simple chaotic dynamics 63

the behavior in the presenceof noise can be determinedby a weighted averageover nearbydeterministic dynamics with a distribution in r of standarddeviationa-0. For example, uniform averagingover a fixed range in r yields a good approximation to observed effects of fluctuations. To obtain thebifurcation diagram for fig. 7 from that of fig. 1, the averaging should use a window in r several standarddeviations wide. Specifically, the standarddeviation for parameteraveragingin this caseshould bea-0 8 x llT~.Appendix C discussesthe useof this rule in moredetail.

This rule provides a simple explanation of the bifurcation gap schematically shown in fig. 9. In thedeterministic limit, for parameters close to r~,it is possible to resolve a periodic orbit or a chaoticattractor of arbitrarily high period. If noise is added, however, the higher periods become unresolvable.According to the rule, this is due to the noise averaging over adjacenthigher and lower periodattractors. In particular, the transition to chaos lowers with increasing noise level as the behavior isaveraged over a wider range of parameters so that periodic orbits at successively lower parameters areaveraged with chaotic attractors.

Above r~,the averagingovernearby parameters smears out all windows of periodic behavior,exceptthe large period 3 window. As has been observed by Mayer-Kress and Haken [27],the effect of noise onthe asymptotic probability density of this particular orbit is quite different thanthe effect of noise on(say) the period 4 orbit in the main period-doubling sequence. The probability density for the period 4orbit consists of four delta functions in the deterministic limit. As fluctuations are added to the system,these delta functions broaden, with essentially zero density in between them. As more noise is added,this process continues until eventually the peaks merge pairwise so that the period of four is lost,leaving only two bands. The behavior of the period 3 orbit is quite different, as shown in the noisebifurcation diagram of fig. 11. As noise is added, in addition to some thickening of the three deltafunctions, a broad background suddenly fills in between the peaks. The sequenceof probabilitydistributions of fig. 12 illustrate the noise-induced bifurcations in more detail. With further increases innoise level, this background rises until the peaks are eventually washed out, as seen in fig. 12c. Thechanges in structure of P(x)are also reflected in the characteristic exponent versus noise level curve offig. 13. The rapid rise of A for small noise corresponds to the broadening of the delta functions of the

0.0 0~ 0.001Fig. II. Noisebifurcation diagramat superstableperiod3 orbit: attractorversusnoiselevel ci at fixed parameterr3 = 3.831874....500 iteratesofeq. (3-I) areplotted afteran initial 500 iterationsateachof the750 incrementsin ci.

64 fl’ C’rutchfield et al., Fluctuations and simple chaotic dynamics

130(a)

lnP(x)

2013.0

(b)

In ~x)

0.0

xInP(x)

20 -5.00 X I 0.0 0.001

Fig. 12. Logplot of (unnormalized)probability densityat thesuperst- Fig. 13. CharacteristicexponentA at thesuperstableperiod3 orbit asaableperiod3orbit, r~= 3.831874... , for threedifferentnoiselevels:(a) functionof noiselevelci. At eachnoiselevel, A wascalculatedwith 106ci=~ i0~,(b) ci= iO~and (c) ci= 102. 106 iteratesof eq. (3.1) were iterationsofeq.(3-1)usingeq.(2-11).TherapidriseinAforverysmallcipartitionedinto 10’ bins. (b) showstheprobabilitydensityasseenin the correspondsto the broadeningof thedelta functions of theperiodicbifurcationdiagramof fig. 7. The noiseinducesabifurcationfrom the orbit, while thenearlylinearincreasein A formoderateci correspondstoperiodthreeorbit to asinglebandatalower noiselevelthan(say)in the theriseof thesingleband“floor” in P(x)seenin theprevioussequence2~cascade,asit is closer, in thesenseofequivalentparametricnoise,to of figures.thesingle largechaoticbandat slightly lower parameter.

deterministicperiod 3 orbit, while for larger a- the nearly linear increasein A reflects the rise of thebackground “floor” in P(x) seen in figures 11 and 12.

To understand this effect in terms of equivalent parametric noise, notice that the period 3 orbit iscloseto achaoticattractorthat is asinglebandaswide asthe period3 orbit itself. At asufficiently largenoise level this chaotic attractoris averagedwith the period 3 orbit, thereby creating the broadbackgroundin the probabilitydensity.

J.P. Crutchfie!deta!., Fluctuations and simple chaotic dynamics 65

In contrast to the sensitivity of periodic behavior to added noise, we find that within variousparameterregimeschaoticbehavioris relatively insensitiveto noiseover a wide rangeof noiselevels.As an example of this, consider the single bandattractorat r = 3.7. Figure 14 contains probabilitydistributions showing the attractor at r = 3.7 in the presence of noise at two different levels. Althoughincreased noise obliterates the detailed delta-function structure in the distributions, the gross features,such as the width and average height of P(x), are unchanged. As a quantitative measure of thisinsensitivity, fig. 15 shows the characteristic exponent as a function of noise at r = 3.7. Over a range innoise level similar to the previous figure, there is essentially no change in the characteristic exponent.This behavior is found for many chaotic regimes, although the corresponding range of noise levelsdepends on the particular band structure. As long as the noise affects only the delta-function structure,the characteristic exponent will change little, but once the noise starts to widen the chaotic bands theexponent will change noticeably.

10.0(a)

In~x) 0.8

400.0 >,.

(b)

InP(x) 02

0.0 o- 0.001Fig. 15. CharacteristicexponentA for thesingle bandat r = 3.7 as afunctionof noiselevelci. At eachnoiselevelA wascalculatedwith 106iterationsofeq.(3-1)usingeq. (2-11).Asthefigure shows,noiseaffectsA

4.Oo______ very little at chaoticparametervaluesover wide rangesof noiselevel.X This dependson the degreeto which noisechangesthe probability

Fig. 14. Logarithmicplotof (unnormalized)probabilitydensityfor the density (say)by wideningit orby filling gaps.For the caseof a singlesinglebandatr = 3.7 andtwo noiselevels:(a)ci = iO~and(b) ci = iO’. band,noiseobliteratesthepeakstructure,but this doesnot contributeThe noiseobliteratesthedetailedstructureof thedeltafunctions,but significantlyto thevalueof A, andsoA is relativelyindependentof noiseotherwisedoesnot changethewidth or height of P(x). level.

Wecan extend the equivalent parametric noise rule to describe the noise-induced changes in A(r).Ratherthan averagingattractorsat adjacentparametervalues,we simply averagethe value of thecharacteristic exponent at nearby parameters. The deterministic A(r) curveof fig. 3 may be smoothedwith a Gaussiandistributionof standarddeviationa-0 — 8 x iO~to obtainan approximation to the A(r)curve of fig. 10. The result of this is shown in fig. 16.

The fractal dimensionof the A (r) curve, mentionedat the endof the previoussection,provided a

66 J.P. Crutchfield et al., Fluctuations and simple chaotic dynamics

Fig. 16. Gaussianaveragingof deterministicA(r) iftfig. 3 over datapointswithin two standarddeviations.Theequivalentparametricnoisestandarddeviation ci~wascomputedfrom eq. (3-12b); i.e., cig = 8ci~= 0.008. The averagingof A(r) with equivalentparametricnoisegives good qualitativeagreement.

roughmeasureof the sensitivity of the self-similar dynamicsto the addition of noise.In light of ourdiscussionof the approximateequivalencebetweenadditiveandparametricnoise,we expectthat at agiven additive noise level o~.periodic windows in the chaotic regimeof size cr0 or smaller will beunobservable. Thus, in the sense that the fractal dimension of A (r) quantifies the repetition ofbifurcationstructureon successivelysmallerscales,it is alsoa measureof howthis detail is washedoutat a given noiselevel. As a first approximation,the noiselevel a- at which we can no longer observeawindow of parameterwidth w is simply proportional to w. If we associatethis noisewidth with the scaleof resolution L~sr s used to derive eq. (2-17), then the length 1 = N L~r s of the A(r) curve at a givennoise level a- is

o~°~’~= a-°6°. (3-13)

We see, then, that the fractal dimension does yield a qualitative measure of the effect of noise on thedetailed structure found in A(r) in the chaotic regime. Before concluding this section, we should alsomention a different type of chaoticbehaviorwhich is encounteredwhenevera tangentbifurcationoccurs. This new phenomenon, which was originally studiedin the context of the Lorenz model byYorke and Yorke [28] and Pomeauand Manneville [291,is characterizedby intermittency,or theappearanceof noisybursts in betweenlaminaror periodicsequences.In what follows, we will considerit in the context of tangentbifurcationsin systemsexhibiting period doubling behavior [31—331.

Consider the third iterate of the map as given by

F~3~(x)= F(F(F(x))) (3-14)

with F = Rx(x— 1) and R = R~= 1 + \/8. For this value of R the map is just tangentto the line x atx~= (0.160,0.514,0.956). For R > R~,F°~(x)passesthroughtheline x giving rise to six newfixed points ofwhich threearestable.Thisphenomenonof tangentbifurcation is responsiblefor the way in which theperiodicwindowsappearin thechaoticregime[14].Now considereq. (3-14)for Rslightly smallerthanR~.A sequenceof third iteratesthen would generateboth a laminarphase(the iteratesmoveslowly in astaircasefashioncloseto anyof thex~,values)andachaoticburstastheiteratesmoveerraticallyunderthemapbeforebeingreinjectedinto anyof thex~values.

J.P. Crutchfieldeta!., Fluctuations and simple chaotic dynamics 67

A number of predictions can be made on the basis of this picture. Expanding F°~(x,R) about Xcand R~one has

F°~(x,R) x~+ (x — x~)+a~(x— x~)2+ b~(R~— R). (3-15)

Settingy~= (x~— x~)/b~the recursion relation for the three-fold iteratetakesthe form

Yn+lYn+aY2n+� (316)

with � = R~— R. For the logistic map a = 68.5. Since the basicstep size in the passage near x~is smallthe recursionrelation nearX~is well describedby the differential equation

dyldl = ay2+ �. . (3-17)

Integrating this equation for � >0 givesfor thenumberof stepsbetweeny~,1and Yout,

l(Yout, yj~,)= ç~Lr[tan_1(yout/V�/a)— tanl(y1n/\/�/a)]. (3-18)

To find the averagelength of alaminarregion, y1,, is averaged over the probabilityP~~(y1~)of enteringsomeacceptanceregion (— G, G) and Yout is set equalto G. So long as G~‘ V~, it is given by

= tanl(G/V�/a), (3-19)

a resultobtainedif the probability distributionP1~(y~~)is takento be uniform over (—G, G).In order to look for this intermittent routeto chaosin the presenceof externalnoise,one hasto

study

= RX~(1— X~)+O~n. (320)

Here~,, is a Gaussianrandomvariablewith (~4) = 5,,~.. Proceeding as before, the 3-fold iteratecan berepresentednearthe contactpointby the Lagevin equation

dy/dl = ay2+ � + a-c. (3-21)

Here~(l) is a Gaussianwhite noisesourcesuchthat

(~(l)e(1’)) = 5(1 — 1’) (3-22)

and a- is proportionalto o~.Introducingthe correspondingFokker—Planckequation,onecan solve forthe averagepath length in the presenceof noise[31].Herewe considerthe scalinglimit valid for small�. In this limit it can be shownthat the averagepathlength in the presenceof noisesatisfiesthescalingrelation

(1) = ~=f(a-2/�3’2). (3-23)

68 J.P. Crutchfie!d eta!., Fluctuations andsimple chaotic dynamics

As u2/�312 vanishes, f goesto a constantand we recover the previousresult,eq. (3-19). Fora-2/�3~’2>1,f(x)— x”3 so that in the limit of large noise

(1)— 1/a-2’3 (3-24)

and a chaotic burst as the iterates were erraticallyunderthe mapbeforebeingreinjectedinto anyof theX~values.

In this section we have described severalimportanteffectsthat additivenoisehason the dynamicsofthe quadratic1D map. We havealso indicatedthat thesecan be understoodin termsof averagingthedynamics over nearby parameterswith an equivalentparametricnoiseandthat the fractaldimensionofA(r) gives a qualitative measure of the sensitivity of the bifurcation featuresto noise.In the nextsectionwe shallconsidera morequantitativedescriptionof thescalingbehaviorof noisenearthe accumulationpoints of period-doubling bifurcations.

4. Noiseas a disorderingfield

Several features of the A(r) curveshown in fig. 10 allow the definition of a criterion for the loweringof the chaotic threshold. First, as any amount of noise obliterates at some scale the detailedself-similaritynearr~,the first positive transition of A near,~becomeseasier to detect. The transition tochaos(fig. 10) is not only lower thanin the deterministiclimit (fig. 3), but the slopeof theenvelopeofA (r) at the zero-crossingis no longer infinite. Second,the pointsof neutralstability (A — 0) correspond-ing to the period-doublingbifurcationsbelow r~,are mademorestableby the addition of fluctuations.In fig. 10, for example,the A (r) curve is smoothedandmademorenegative,indicating greaterstabilityat the pointsof bifurcation. Thesetwo featurestakentogethermakethe first positivetransition of Aunambiguous.Hence,wecan definethe onsetof chaos,r~(o-),as the valueof the parameterr at noiselevel a- for which A (r) first becomespositive. For conveniencewe will usethe normalizedparameterF = (r — r~)/r~.Froman extensivestudyof A(r) versusnoiselevel a-, in conjunctionwith the correspond-ing bifurcation diagrams,wecan summarizethe effect of fluctuationson the cascadebifurcation in fig.17, which avoidscertainambiguitiesassociatedwith fig. 9. Thus wecan nowfollow differing bifurcationpathsto the samepoint in the diagramof fig. 17 unambiguouslyand without the needfor a specialtransitionrule as is the casefor fig. 9.

Onefeaturethat becomesapparentin the diagramof fig. 17 is the scalingbehaviorof the onsetofchaos r~(a-)as a function of noise level a-. If we denoteby r~(a-)the value of r for which thecharacteristicexponentbecomespositiveat fixed noiselevel, we can write

k0a-

1’ (41)

wherey is a critical exponentsummarizingthe power law behaviorof the onsetof chaosas a functionof noise level and k

0 is a constant of proportionality depending on the periodicity of thecascade.Usingeq. (2-11), we have determinedthe exponenty and constantk0 from numericalexperiments to be

y=O.82±0.Ol (4-2)

J.P. Crutchfield eta!., Fluctuations and simple chaoticdynamics 69

0 a- 2x10 -35.0 In ~ -5.0Fig.17. A renditionofthebifurcation “gap” whichdoesnothaveanyof Fig. 18. Loweringofthechaoticthresholdr~with increasingnoiseleveltheassociatedambiguitiesof fig. 9. Periodp andbifurcationvaluesof the ci forthe2” cascadeon alog—logscale.Theabsolutevalueofr~is usedasparameterversuslogarithm of the noise level ci. The dashedline r~<0. The10 ordersof magnitudein r~correspondto orbitsof periodsrepresentsr~(ci) (definedin text)versusci andindicatesthathighernoise from2~to 216, in thedeterministiclimit. Thenoiselevelrangesover12levels induceanearliertransitionto chaos.For numericaldetailsofr~(ci) ordersof magnitude.Eachdatapointcomesfrom abinarysearchatfixedrefer to thefollowing figure. Abovethedashedline theattractorsare r of aA vs. ci curvefor thezero-crossingof A.During thesearcheachAchaotic,below periodic. was calculatedwith N = 106 or 10’ (small ci) in eq.(2-li).

for both q = 1 and3, and

k1 = 0.60±0.01, k3 = 0.40±0.01. (4-3)

Figure 18 shows a log—log plot of r~versus a- over 12 ordersof magnitudein a-. The range in rcorresponds,in the deterministiclimit, to orbits from period2~to period 216.

If we recall the definition of a-,. as the noiselevel at which onecanobserveanorbit of at mostperiodp andif we define r~(a-~)= r, then eq. (2-9) written in termsof thenormalizedparametergives

= 5 (4-4)

and,also,

~ = 5. (4-5)

Using eq. (4-1), we then obtain [1]

= S~. (4-6)

Thusfor acascadewhereonecanobserveatmostaperiodof p at a given noise level a-,,, onemustreducethenoiselevel by a factor of 5k” — 6.6in order to resolveaperiod of 2p•

Thebehaviorof A nearr~in theabsenceandpresenceof noiseas shownin figures9 and 17, leadstothe notion of externalfluctuationsacting as a disorderingfield on the deterministicdynamics.As wepointedout in section2, in the deterministiclimit A behavesas adisorderparameterfor chaos,with apowerlaw given by eq. (2-12). As the strengthof theexternalnoiseis increased,A acquiresafinite value

70 JP. C’rutchJleld et a!.. Fluctuations and simple chaotic dynamics

0.05

x

0.00.0 0~ 0.001

Fig. 19. CharacteristicexponentA versusnoiselevel ci at r~for the2” cascade.A was calculatedwith N = 106 in eq. (2-11)for 100 valuesof ci. Thestepscorrespondto thenoise-inducedmergingof bands.The overall powerlaw behavior,though,can be summarizedby thecritical exponentB ofeq. (4-7).

at r~,while the zero crossing is renormalized according to eq. (4-1). Furthermore,sincethepresenceof abifurcationgapimplies the smearingof bandsin the chaoticregime, A will no longer possessan infiniteslopeat the thresholdin the presenceof a disorderingfield.

Theseconsiderationsleadus naturally to expectthat at r~,the Lyapunov characteristicexponentAwill scalewith noiseaccordingto

A(r, s)= Aa-° (4-7)

where 0 is a universalexponentwhich plays a role similar to the isothermalexponentin criticalphenomena[34]. Figure 19 showsthe dependenceof the characteristicexponentat r = r~on the noiselevel a-. In addition to an overallpowerlaw behavior,several“steps”areapparentin thecurveof A (a-).The stepsareseparatedby factorsof S —j 6.6 in the noiselevel andcorrespondto the noise-inducedmerging of bands.The sequenceof probability distributions of fig. 20 illustratesthe rescalingof thedynamicsby changesin the noiselevel of factorsof ~ As the noise increasesby 5~”from (a) to (b)and againfrom (b) to (c), the structureof the band probability distribution is seen to be the same,exceptfor a spatialrescaling.Figure 19 alsoshowsthat A increasesrelatively linearly with noiseleveluntil bandsmerge,at whichpoint A levelsoff with increasingnoiselevel becausetheprobability densitychangeslittle. This is verified by the noisebifurcationdiagramof fig. 21 which shows16 bandsmergingintO 8 and8 bandsmerginginto 4 at noiselevels correspondingto the stepsin A(a-) of fig. 19.

In order to determinethe validity of eq. (4-7) we havemeasuredthe changeof A at r = r~withincreasingnoiselevel over 9 ordersof magnitudein a-. As can be seenin fig. 22, a log—log plot of thedatarevealsa straightline from whichwe can extractthe values

0~0.37±0.01, A10.58±0.01 (4-8a)

for theperiod 1 cascadeand

0 =0.37±0.01, A3= 1.13±0.01 (4-8b)

J.P. Crutchfield eta!., Fluctuations andsimple chaotic dynamics 71

11.0 ~ —

(a)

lnP(x)

.0 —

11.0 —

(b)

lnP(x)

L~11.0

(c)

In P(x)

.00

Fig. 20. Logarithmicplot of (unnormalized)probabilitydensityat theaccumulationpoint r, = 3.569945.. . of the2” cascadefor threedifferent noiselevelsseparatedby factorsof 8’~—6.6: (a) ci = 2.3x i0’, (b) ci = 1.5 x iO~,and (c) ci = iO~.The last correspondsto the noiselevel of fig. 7 inwhich4 bandsareapparent.Thesequenc.eillustratestheband-mergingbifurcationsat r~inducedby increasingnoise levels.Again, 106 iterationsofeq. (3.1)werepartitionedinto 10’ bins. The differencein noiselevels is chosento illustrate thenoisescalingstructureof P(x).

for theperiod 3 cascade.Thedeviationsfrom astraight line, seenas groupingsof 2 and3 datapointsinfig. 22, aredueto the noise-inducedbandsmergingsfound in fig. 19.

Theseresultstogetherwith the Huberman—Rudnickscalingof the characteristicexponent,eq. (2-12),and recentscaling theories[35,361, suggestthe existenceof a homogeneousscaling function F[F, a-]suchthat

A(Pa-)zrr_~—F[F,a-] (4-9)

72 J.P. Crutchfie!deta!., Fluctuations and simple chaotic dynamics

{x~} -

00.0 a- 0.001

Fig. 21. Noisebifurcation diagramdiagramat 2” accumulationpoint: attractor{x~}versusnoise level ci at r~= 3.56995 500 iteratesof eq. (3-1)areplotted afteran initial 500 iterationsat eachof thell)~incrementsin ci./

-10.030.0 no- 0.0

Fig. 22. Log—log plot of A versusnoiselevel ci showingthedatausedto obtain B = 0.37and A1 = 0.58 for eq. (4-7). Thegroupingsof 2 and3 data

points reflect the noise-inducedband mergings.A was calculatedwith N = 106 in eq. (2-11)at 30 values of ci which ranged over 9 orders ofmagnitude.

and satisfying a scaling behavior, i.e.

F[La~p,L°’~a-]= LF[F~ a-], (4-10)

wherea~anda~arescalingparameters.In particular, when a- = 0, eqs. (4-9) and(4-10) imply

A(F, 0) = (~F)°~=~’A(—1,0) (4-11)

which togetherwith eq. (2-12) gives

r= (1— a,~)/a,. (4-12)

J.P. Crutchfie!d eta!., Fluctuations and simple chaotic dynamics 73

Similarly, we can expressthe exponent0 in termsof the scalingparametersar anda~.Setting r = 0 ineqs.(4-9) and (4-10) andletting a-—*0, we obtain

A(0, a-) = a-~”A(0, 1) (4-13)

which, togetherwith eq. (4-7) gives

0 (1—a,~)/a,,.. (4-14)

In order to obtain an explicit relation between T and 0, we can introduce a noise susceptibilityX(F~a-) defined as

X(F~a-) dA/da-. (4-15)

The magnitude of the noisesusceptibilityX gives a measure of the sensitivity of the dynamics to theaddition of noise.For example,at the transitionsto chaosand at points of superstability,X diverges,although nearbyit is finite and positive. Actually, nearthe transition to chaos,the behaviorof X iscomplicatedsomewhatby the accumulationof self-similar structure.The additionof a smallamountofnoise, however,truncatesthis structureso that the natureof the divergenceat the transitioncan bestudiedin practice.Nearr~,then,X shouldalso obeyascaling law

X(F~0) = cF~. (4-16)

Taking derivatives of eqs. (4-9) and (4-10) we obtain

w = (2a~— 1)/a. (4-17)

which together with eqs. (4-12) and (4-14) lead to the relation

w r(01—1). (4-18)

Using the known values for r and 0, eq. (4-18) yields a prediction for the noisesusceptibilitycriticalexponent of w = 0.77. In the previous section we noted that the addition of noise lowered thecharacteristicexponentat the period-doublingbifurcations.Thus X is negative nearperiod-doublingbifurcations and apparently diverges at this point of bifurcation too. We also pointed out that for asingle chaotic band the characteristicexponentis affectedvery little overa wide rangeof noiselevel. Insuchcases,the noisesusceptibilitywill be very small, if not zero.

The results of this section show that the existenceof a homogeneousscaling function F withuniversalpropertiesleadsto scaling relationsvery similar to thoseencounteredin critical phenomena.In particular,one is able to accuratelypredict the value of the exponentsthat relate the effect ofexternal fluctuations on the chaotic behavior of deterministic systemsexhibiting period-doublingcascadebifurcations.Recentwork [35,36] pointsto a renormalizationgroupdescriptionof the scalingbehavior revealed by the above numerical investigations.In fact, these approachesyield criticalexponentsin excellentagreementwith thosereportedabove.

74 J.P. Crutchfieldeta!., Fluctuations and simple chaotic dynamics

5. Concluding remarks

Using the simplest dynamical systemwhich undergoescascadebifurcationswe studiedin aquan-titative mannerthe fluctuation effects reportedearlier in a system of nonlinearordinary differentialequations[1]. In particular,we havedescribedthe fluctuation-inducedgapin the cascadebifurcationsequenceand the scaling behavior of both the threshold noise and the disordering field. Ourinvestigationsprovidestrongevidencefor theconsiderationof the Lyapunovcharacteristicexponentasa disorderparameterfor chaos.We havealsoderivedthe scalingbehaviorof the effectsof fluctuationsusing resultsfrom the universalscalingtheory for the period-doublingbifurcation. Thus theseresultsshouldbe relevantto otherdynamicalsystems,including weakly turbulent fluid flows, which undergocascade bifurcation.

The existence of deterministic models that show chaotic or unpredictable behavior puts in a newlightquestionson the physicalorigins of noiseprocesses.With this in mind, we would like to considerthelarger context in which this papershould be placed.For modelsof classicalphysical systemswe candistinguishthreetypes of fluctuation. The first, observationalnoise,is due to the finite resolution ofphysicalmeasurements,that is, instrumentationerror. Problemsassociatedwith observationalnoiseareoftenconsideredtheprovinceof the mathematicaltheory of communication,which describestheeffectof randomerrorson a signal representingsomephysical quantity [37]. When a physical system is incontactwith a “heat” bath in which alargenumberof particles(or degreesof freedom)areexcited,asecondtypeof fluctuation,externalnoise,appears.This secondtypeof noiseaddsa stochasticforce tothe dynamicalequationsandthe resultingnon-deterministicproblemis solved with statisticalassump-tions and techniques[38].The LangevinequationdescribingBrownian motion exemplifiesthisclass ofstochasticmodel.Finally, chaoticdynamicalsystemsexhibit stochasticityor randombehavior,althoughtheyarecompletelydeterministic.This deterministicrandomnessis the third type of fluctuation,whichwe shall call intrinsic noise.It is of interest,for example,to the studyof nonlineardifferential equationswhosephasespacedescriptionsrequireat leastthreedimensions.

In current attemptsto relatedeterministicchaoticmodelsto turbulentphysicalsystems,one of theoutstandingproblemsis the interactionbetweeneachof thesetypesof fluctuation.On onehand, noiseappearsas an everpresentbut undesirableartifact which, moreoften than not, complicatesexperimen-talists’ interpretation of their data. On the other hand, from the theoretical point of view, theintroductionof fluctuationsin a problemimplies a simplification of a model. Suchan ansatzrepresentsan explicit lower boundon the level of descriptionbelow which the detaileddynamicsare not to beconsidered.The results we havepresentedhere and previously [1] suggesta role for fluctuationsintermediatebetweenthe extremesof experimentalcomplexity and theoreticalsimplicity. We haveshownthat abroadclassof transitionsto chaoticbehaviorcan be characterizedby its alterationin thepresenceof thermal-like noise. In principle, this should aid in distinguishing the type of modelappropriate to describe observed, random behavior.

Aside from the particularbehaviordiscussedherefor the cascadebifurcation, we wish to emphasizethe importanceof consideringfluctuations in modeling turbulent physical systems.From an under-standingof the changesinducedby fluctuationsin the geometryof the phasespaceflow, one maybeable to elucidate the relationshipbetweensimple chaotic dynamicsand turbulent physical systems,such as turbulent fluid flow [39] and noisy solid state systems[1,5]. Currently, simple chaoticmodels serve only as metaphors for turbulent behavior in continuous(infinite dimensional)physicalsystems. Although one finds electronic and mechanicalsystemscorrespondingto chaotic models[40]and recent application of phasespacereconstructiontechniquesto stirred chemicalreactions[41],there

J.P. Crutchfie!deta!., Fluctuations and simple chaotic dynamics 75

is as yet no direct experimentalevidencethat chaoticdynamicsdescribesobservedrandombehaviorincontinuumsystems,such as turbulentfluid flow. At present,the comparisonof bifurcationsequencesofpowerspectrabetweenexperimentandmodel systemsandthe reconstructionof phasespacepicturesprovide the only methods of validation of the conjecture. Hopefully, an understanding of therelationship between the three types of fluctuation (observational, external and intrinsic) in modelsystems will lead to experimental tests which will determine the relevance of chaotic dynamics to noisyphysical processes.

We should mention another approach to distinguishing between types of intrinsic noise that is ofinterest in distinguishing between intrinsic and thermal noise. It considers, first of all, the reconstructionof phase space dynamics from a single experimental variable and then determinesthe number ofindependentvariables underlying the reconstructed dynamics, that is the intrinsic dimensionality of aphasespacedescription[42,43]. According to this approach,different typesof intrinsic noiserequiredifferent numbersof phasespacedimensions.Thermal noise, consideredas a deterministicprocess,would be characterizedby a relatively large numberof phasespacedimensions.It is still an openquestionwhetherpracticalalgorithmscan bedevisedto distinguishbetweenthermalandintrinsic noisewhen the dimensionof the underlyingphysical processis inherentlylarge. At the presenttime, thereappearto be substantialcomputationaldifficulties for the experimentaldeterminationof the intrinsicdimension of physical processes even when the dimensionis as low as 5 (say), not to mention theproblems of visualizing a chaotic attractor of that dimension with the reconstruction techniquescurrently proposed.The utility of thesetechniqueswill probably be limited in answeringquestionsabout the interaction of thermal noise and chaotic dynamics. Nonetheless, the conceptual frameworkthat thesetechniquesprovide allows one to understandthe transition from low dimensionalchaoticdynamicsto high dimensional,deterministic,thermal-likenoiseprocesses.

From a different perspective,Ruelle [44] and Shaw [19] estimate that the time necessary for athermalfluctuationto affect themacroscopicmotion of fully-developedturbulencein afluid is relativelyshort, being on the order of secondsfor air. Ruelle concludes,however,that in this regime, becausethermal fluctuationsmust competewith many other perturbationsof similar energywhich are alsoamplified by the flow, changes in the level of thermal fluctuations would probably not be experimentallyobservable. In the weakly turbulent regime though, the changesin the level of thermal fluctuationscould be quite noticeable, as suggested by our results on the cascade bifurcation to chaos. Fluctuationsare of interest from a mathematicalpoint of view, thermal fluctuationsmay also play an importantrole in selecting the relevant stationary measure on the chaotic attractor describing turbulence. Inprinciple,therearemanysuchmeasures,but, as Kifer [45]hasshownfor Axiom-A systems,only oneisstableundersmall stochasticperturbations.

The picture of microscopicfluctuationsdeterminingmacroscopicbehavior is one that is generallyassociatedwith locally unstableor chaoticdynamics.Shaw [19] developsthis notion by consideringchaotic dynamical systems as sources of information; this information originatesin the microscalesbeyond experimental resolution. He discusses the unpredictability of chaoticsystemsin termsof finitemeasurementresolution,or observationalnoise, to usethe aboveterminology. As an example, heestimates how long it takesa chaoticsystemto becomeunpredictable.If the stateof a systemcan bedetermined to within somefinite resolution a- (measuredrelative to the total numberof resolvablestates) and if one knows a priori the information loss rate A0 (such as the maximum Lyapunovcharacteristicexponent),then the system is effectively unpredictablea time t = ln(cr)/Ao after ameasurement. As a first approximation to an observation theory of chaos, this argument raises aninterestingquestionfor experimentalists:Without knowing the rate of information loss, how can the

76 J.P. (‘rutchfie!d eta!., Fluctuations and simple chaotic dynamics

rate itself be measured experimentally in the presence of observational noise? Or, indeed, in thepresence of the other types of noise mentioned above? [461.

As an explicit example of the relationship between thermal-like fluctuations and intrinsic noise, wehave studied a model exhibiting chaotic behavior, eq. (3-1), which included a stochastic force. This“thermal fluctuation” term was in practice a deterministic pseudo-random number generator im-plementedon a digital computer.Its iterative algorithmwas operatedon a different time scalethantheID map in order to let the correlationsdie out and so give the desiredstatistics.This time scale isdetermined by the degree of randomness of the algorithm which can be measured with a characteristicexponent,that is, a measureof the divergenceof nearbystates.The introductionof pseudo-randomnoise imposed a scale of resolution below which we did not consider the dynamics. This “thermal” noisewas characterized by suitable statistical quantities, such as the mean and standard deviation, although itwas a deterministicprocess.From the oppositeperspective,this suggests that in pursuing the under-standingof a noisyphysicalprocessto finer degreesof resolution,the “noise source”may appearas adeterministicnonlinearsystemwith chaotic dynamics.This certainly was the casefor our numericalexperiments.

Acknowledgements

The authors have benefitedfrom discussionswith JohnGuckenheimer,Michael Nauenberg,NormanPackard,JoeRudnickand Rob Shaw.They also wish to thank Ralph Abrahamand JoeRudnick fortheir support of the authors’ computationalwork at UCSC andto acknowledgethe help providedbythe late Joe Maleson with the computers at Xerox PARC. Rob Shaw has generously provided fig. 2from his unpublished work. This work was supported in part by the National ScienceFoundationGrantNo. 443150-21299.JDF gratefully acknowledges the support of the Fannie and John Hertz Foundation.JPCwas supported by a University of California Regents Fellowship.

Appendix A. Fluctuations in a driven anharmonic oscillator

Although characteristic exponents provide an unambiguouscriterion for the earlytransition to chaosinducedby increasingnoiselevels,for systemsmorecomplexthan 1D maps,their calculationbecomesquite time consuming. In the caseof the driven anharmonicoscillator initially studied[1] one canobserve this effect in the variation of Poincaré sections with noiselevel more readily, in fact, than inchanges in the 1D mapattractorswith noiselevel. Generally,chaoticbehaviorappearswith a particulardegreeof mapping,or folding, of orbitsonto eachother.Calculationof characteristicexponentsgivesthebestdeterminationof when this folding occursin 1D maps.For driven oscillators,and otherchaoticsystemswhoseattractorsappearsheet-like(of topologicaldimensiontwo), on the otherhand,thispointis reachedwhen the first folding appearsin the attractor’sgeometry, as revealedin a sequenceofPoincarésections.From the underlyinggeometryof the attractoronecan then infer the transitiontochaos.

We includeheretwo setsof Poincarésections(figures Al andA2), takenat differentnoiselevels forthe oscillator of ref. [1], to illustrate in another context the qualitativeeffectsof fluctuationsdiscussedinthis paper. The anharmonic oscillator studiedthereis given by

I+g~+ax—bx3=Fcos(wt). (A-i)

J.P. Crutchfie!d et a!., Fluctuations and simple chaotic dynamics 77

~ (e} / (d) / / (c) . (b)

Fig. Al. A setof phase-zeroPoincarésectionsat different driving frequenciesalongthe horizontalaxis for theanharmonicoscillatorstudiedin ref.[1] for a (nornalized) noiselevel a= iO~.The set (a)—(f) showsa cascadebifurcation asthedriving frequencyis lowered (toward the left in thefigure). At this noiselevel only a maximumperiodof four can be observed.The folding geometryis apparentfor the bandattractorsin (d)—(f).

(d) (c) (b) (a)

/ j../ /

Fig.A2. Thesetof Poincarésections(a)—(d) aresimilar to thefirst setin figure Al, exceptthey aretakenat anoiselevel ci = 10_2. Only a maximumperiodof two is presentfor this noiselevel. Noticealsothat thefluctuationsspreadmore in thedirectionalongtheunstablemanifold(bands)thanalongthestablemanifold (transverseto thebands).

Figures9 and17 summarizetheobservableperiodsat agiven noiselevel andperiodicity, as is seeninfigures Al and A2. An important featurerevealedby the Poincarésectionsis that the fluctuationsspreadout the orbits morealong the “soft” directionswithin the attractor(the unstablemanifold) thanin the transversedirections (stable manifold), along which orbits contract very rapidly onto theattractor.The explanationfor the inducedtransition to chaoswith increasingnoiselevel, then,is thatthe fluctuations fill out the folds of the attractor, causingthe orbits to describea geometrywhich iseffectively chaotic. The existenceof such folding geometry cannot be easily inferred from thebifurcationdiagramsof the 1D map.

Appendix B. Characteristic exponentsin period-doubling regimes

The characteristicexponentin periodic regimes,as seen in fig. 3, hasa distinctive shapebetweenperiod-doublingbifurcations. Although the dependenceof the characteristicexponent A on theparameterr cannotbe analytically calculatedin general,it can be straightforwardlydeterminedfromeq. (2-10) for a few simple cases.The nature of the divergenceat superstableorbits can also beelucidatedusing eq. (2-10). Furthermore,oncewe havetheseresults,a scaling of A(r) in the periodicregime,analogousto the scalingabover~of eq. (2-12),follows easily.

A period p orbit consistsof the set of stablefixed points~x1},i = 1,. . . , p, of the pth iterateof themap.Thesearegiven by the equation

x=f”(r,x~). (B-i)

Eq. (B-i) implicitly determinesthe dependenceof the fixed points x1 on r. Thus,the bifurcation ofperiodic orbits reducesto the studyof the real rootsx, of the polynomial in r, x~— f”(r, x.). Only p oftheseroots correspondto the pointsof the stableorbit; the otherscorrespondto unstableorbits. Forperiod 1 orbitsof the logistic equation,we havethe roots x = 0 and

78 J.P. Crutchfield et al., Fluctuations and simple chaotic dynamics

x1=(r—l)/r. (B-2)

The root x1 describesthe stableperiod 1 orbit for r in [1,3]. For period 2, therearefour rootsof thepolynomial

x —f(r, x) = x — r2x(1 — x)(l — rx(1 — x)). (B-3)

The period 1 root andx = 0 are alsoroots of this polynomial, but they correspondto unstableorbits.The remainingtwo roots describethe stableperiod 2 orbit for r in [3,r

2_4]; theyare

~±= 1+r±(r22~3)1~’2~ (B-4)2r

Generally,a periodp orbit is determinedby the roots of a polynomialof order2”. For aperiod2~orbit,the roots corresponding to the unstable orbits of periods ~ ~ and1, can be factoredout ofthe polynomial x — f”(r, x), in principle. This would leave a 2~-orderpolynomial whose roots arethepointsof the stable2” orbit.

From the equationsfor the dependenceof the periodic orbits on r, the dependenceof thecharacteristicexponentcan beobtainedfrom eq. (2-10).For aperiodp orbit {x,}, the probabilitydensityP(x) is a set of p deltafunctions, cS(x — xe). In this case,eq. (2-10) becomes

A(r) = ~ ln~f’(r,x~)I. (B-5)

For the logisticequationwe havestudied,the slopeis given by

f’(r, x~)= r(i — 2x,). (B-6)

For the period 1 orbit, then,the characteristicexponentis

A(r) = 1n12— rI, (B-7)

for r in [1, 3]. The argumentof the logarithmindicatesthat the bifurcations,whereA —~0, from x = 0 tothe period 1 occursat r = 1 and from period 1 to period 2 occursat r = 3. It also shows that thesuperstableorbit, whereA —~ —oc, is found at r = 2. Similarly for the period 2 orbit, we find

A(r)= lnlr2— 2r—41, (B-8)

for r in [3, r2_41. The argumentof the logarithmyields r2_4= 1+ \/6 andfor the superstableperiod 2

orbit r= i+V5.The characteristicexponentdivergesat superstableorbits {x,} as oneof the x, approachthe critical

point x~= 0.5, wherethe slope vanishes.To show this, in general,we must first determinehow theparticular x in question,denotedby x *, approachesx,, as the parameterapproachesthe superstablevaluer~.x” is given by onebranchof roots of the polynomial x —~(r,x). This gives x” implicitly as afunction of r; that is, if g(r) is the single branchcontainingXc, then wewrite

J.P. Crutchfield a al., Fluctuations and simple chaotic dynamics 79

x*(r)=g(r) (B-9)

wherex~= g(rs). As can beseenfrom thebifurcation diagramof fig. 1, g(r) is a smooth curvenearr~.Ifwedefinedr = r — r~,then

dx*(r) = ~- g(r) dr. (B-b)

Closeto r~,the slopeof g(r) is very nearlya constantk = (d/dr)g(r~),whichgives

dx*~~~kdr. (B-il)

Next, we must determinehow the slope changesnearr~.If we define dx = x* — x~and dr as above,

simpleexpansiongives

f’(r, x *) = f’(r5, x~+ dx) —~f’(r~,x~)+ f’(r~,x~)dx + O(d2x). (B- 12)

By definition the first, third andhigher, termsvanish.Furthermore,f(r, x) = —2r so that we have

f(r, x*) = —2r~dx (B-13)

nearr~.Fromeqs. (B-li) and(B-13), the slope’sdependenceon r near r. is thengiven by

f’(r, x*).~~—2r5k dr. (B-14)

Thedivergenceof A (r) is dominatedby the termin eq. (B-5) whoseslopeis vanishing,andsowe canignore the contributionsfrom the other(p — 1) pointsalongthe orbit. Thus,we seefrom eq. (B-5) thatthe divergenceof A(r) is logarithmic nearr~andgiven by

A(r)-’-- lnl2r~kdr~—~lnIr~—rj . (B-15)

We now turn to the scalingpropertiesof the superstabledips. As r approachesr~the width w,, of thenth dip in A (r) decreasesexponentiallyatarategiven by

w,, = r~±1— r,, -= ö~”. (B-16)

Furthermore,astheperioddoublesfor eachsuccessivedip, themagnitudeof A (r) decreases by a factor oftwo from its valueon the previousdip. Within asingle dip then, we can write

A~(r)=~ln 2(r~—r) , (B-17)2 w,,

whereA0 is a constantand r~is the valueof the parameterr at the superstableorbit. The depthof thesuperstabledips, as seenin fig. 3, decreasesas r approachesr~,although in principle, A(r) is infinite atthe superstableorbits.Thiseffect is due to theresolutionin r at which fig. 3 was made,but alsoreveals

80 J.P. Crutchfie!d a a!., Fluctuations and simple chaotic dynamics

a scalingbehaviorof the superstabledips.This behaviorcan beaccountedfor by calculatingtheaverageof A(r) betweenperiod-doublingbifurcations.This averageis givenby

A~=-~--J~ln~2(Ts~ dr. (B-18)

Since the dips are nearly symmetric, the lower limit can be changed from r,, to r~.Performing theintegration we find

A~= —A0/2”. (B-l9)

Solving eq. (2-9) for n in termsof r, we find

A(r) = —A~(r— r~)1 (B-20)

whereT = In(2)/ln(~) = 0.4498...,and A~is a constant. Thus, we see that the averagevalue of A(r)scales in a manner analogous to the envelope of positive characteristicexponentabover~,as shown inref. [22].

Admittedly thesearesimple considerations.A moredetailedanalysisalongtheselines,as developedby Daido [47],shows that the argument of the logarithm in eq. (B-i7) approaches a universal polynomial.Eqs. (B-7) and (B-8) are the first approximationsto the universalexpressionfor the Lyapunovexponentin the period doubling regime.

Appendix C. Equivalenceof parametric and additive noise

The equivalent parametricnoise rule, introduced in section 3, allows one to construct a goodapproximationto the noisybifurcationdiagramof fig. 7 from the deterministicbifurcationdiagram offig. 1. Simply stated,the noisybifurcation diagramis obtainedby the convolution of the deterministicbifurcation diagram with a Gaussianprobability distributionin r whose standard deviation is given byeq. (3-12b).A goodapproximationof this processcanbeconstructedin thefollowing manner:In apieceof paper cut a slit that is parallel to the x axis of fig. 1. The width of the slit in the r direction should beequal to several standard deviations of the equivalent parametricfluctuations.To estimatethe noisyattractor at any particular value, place the slit over fig. 1 so that its midline lies at the parametervalueof interest. Now project all the pointsof the neighboringattractorsthat arevisible within the slit ontothe midline. If the width of the slit is appropriately chosen, the resultingdistributionof pointswill givethe attractor in the presence of noise.

As a first estimateof the properslit width, we assumethat the slit width is constantin x anduseeq.(3-12b). If the slit width is set at 3a-q, then 99.9% of the equivalentparametricfluctuationswill havemagnitudes that lie within the slit. For the example of fig. 7 the magnitudeof the additive fluctuationsisa-,.. = iO~andthe estimatedslit width is 3a-q= 24a-~ = 0.024.That is, the slit width to be used with fig. Ito obtain fig. 7 should be 2.4% of the horizontal axis.

Let us now construct an approximation of fig. 7 usingsuch a slit andfig. 1. Notice that the slopes ofthe bifurcationcurvesin fig. 1 arelarge nearthe period-doublingbifurcation to period 4. Consequently,the noisy bifurcation curves of fig. 7 are broadened near this point by the projectionontothe midlineof

J.P. Crutchfield eta!., Fluctuations and simple chaotic dynamics 81

the slit. Similarly, the bifurcation to the period 8 orbit is so close to the period 16 that the values of xvisible within the slit neverseparate.The period 8 orbit, and all higher periodorbits, neverbecomevisible in the noisy bifurcation diagram of fig. 7. The other noise effects discussed in section 3 can beexplainedin an analogousmanner.

This approximationneglectsthe x-dependenceof the equivalentparametricfluctuations.Notice, forexample,that the width of the upperfork of the period 2 bifurcationcurve is muchthinner thanthat ofthe lower fork. One of the reasonsfor thisis apparentfrom the slit construction:the upperbifurcationcurvehas a smaller slope. Another reason is that the equivalent fluctuations arenot ergodic.Eq. (3-6)predictsthat the q,, fluctuations are larger when x,. — 0.8, i.e. the upper fork, than they are when

0.5, i.e. the lower fork. At first glancethis may appearto havean effect counterto that which wearetrying to explain. However,the influenceof the largerfluctuationson the upperfork is only felt onthe succeedingiteration, that is, by the lower fork. Both of theseeffectscombineto makethe upperbranch for fig. 7 considerably narrower than the lower.

In order to take this second effect into account automatically,ratherthan usinga slit of fixed width,the width can be varied as a function of x. To do this it is necessaryto considerthe amplitude of theparametricfluctuationsas a function of the value of x on the next iteration, when they are moststrongly felt. From eq. (3-5) we see that

p~=q~f(x~) (C-I)

andthat

f(x~)=(x~+1—p~)/r. (C-2)

Assuming p,, ~ 1 and eliminating f(x~)from eqs. (C-i) and (C-2), we find

q,. — rp,,Jx~÷i. (C-3)

The fluctuation q,~ is felt on the (n + 1)st iteration; that is, q~affects X~+i.Thus, atanygiven valueof x

the slit width w(x) is given byw(x)= 3a-q(x)= 3ra-,.,/x, (C-4)

if onetakesthreestandarddeviations.

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82 J.P. Crutchfie!d eta!., Fluctuations and simple chaotic dynamics

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